quasitriangular bialgebra



A quasi-triangular bialgebra / triangular bialgebra is a bialgebra equipped with just the right structure such as to make its category of modules into a braided monoidal category/symmetric monoidal category.


Let AA be an algebra in a symmetric monoidal category CC with symmetry τ\tau; fix m,lm,l and DA kD\in A^{\otimes k} and let 1i rl1\leq i_r\leq l for 1rm1\leq r\leq m be different. Then denote D i 1,,i mA nD_{i_1,\ldots,i_m}\in A^{\otimes n} as the image of R1 (lm)R\otimes 1^{\otimes (l-m)} under the permutation which is the composition of the mm transpositions (r,i r)(r,i_r) of tensor factors interchanging rr and i ri_r. In the following CC is the monoidal category of kk-vector spaces.

A kk-bialgebra (in particular kk-Hopf algebra) is quasitriangular if there is an invertible element RHHR\in H\otimes H such that for any hHh\in H

τΔ(h)=RΔ(h)R 1 \tau\circ\Delta(h) = R\Delta(h)R^{-1}

where τ=τ H,H:HHHH\tau=\tau_{H,H}:H\otimes H\to H\otimes H and

(Δid)(R)=R 13R 23 (\Delta\otimes id)(R)=R_{13} R_{23}
(idΔ)(R)=R 13R 12 (id\otimes\Delta)(R)=R_{13} R_{12}

An invertible element RR satisfying these 3 properties is called the universal RR-element. As a corollary

(ϵid)R=1,(idϵ)R=id (\epsilon\otimes id) R = 1,\,\,\,\,\,(id\otimes\epsilon)R = id

and the quantum Yang-Baxter equation holds in the form

R 12R 13R 23=R 23R 13R 12 R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}

A quasitriangular HH is called triangular if R 21:=τ(R)=R 1R_{21}:=\tau(R) = R^{-1}.

The category of representations of a quasitrianguar bialgebra is a braided monoidal category. If RR is a universal RR-element, then R 21 1R_{21}^{-1} is as well. If HH is quasitriangular, H copH^{cop} and H opH_{op} are as well, with the universal RR-element being R 21R_{21}, or instead, R 12 1R_{12}^{-1}. Any twisting of a quasitriangular bialgebra by a bialgebra 2-cocycle twists the universal RR-element as well; hence the twisted bialgebra is again quasitriangular. Often the RR-element does not exist as an element in HHH\otimes H but rather in some completion of the tensor square; we say that HH is essentially quasitriangular, this is true for quantized enveloping algebras U q(G)U_q(G) in the rational form. The famous Sweedler’s Hopf algebra has a 1-parameter family of universal RR-matrices.


Tannaka duality

A quasitriangular structure on a bialgebra corresponds to a braided monoidal category structure on the category of modules of the underlying algebra. (For instance chapter 1, section 2 of (Carroll)).

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module


  • V. G. Drinfel’d, Quantum groups, Proc. ICM 1986, Vol. 1, 2 798–820, AMS 1987.

  • S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Int. j. mod. physics A, 5, 01, pp. 1-91 (1990) doi:10.1142/S0217751X90000027

  • S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.

  • A. U. Klymik, K. Schmuedgen, Quantum groups and their representations, Springer 1997.

  • V. Chari, A. Pressley, A guide to quantum groups, Camb. Univ. Press 1994

  • Robert Carroll, Calculus revisited

Last revised on September 2, 2013 at 15:35:57. See the history of this page for a list of all contributions to it.